Highest averages method
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The highest averages method or divisor method is the name for a variety of ways to allocate seats proportionally for representative assemblies with party list voting systems. It requires the number of votes for each party to be divided successively by a series of divisors. This produces a table of quotients, or averages, with a row for each divisor and a column for each party. The nth seat is allocated to the party whose column contains the nth largest entry in this table, up to the total number of seats available.^{[1]}
An alternative to this method is the largest remainder method, which uses a minimum quota which can be calculated in a number of ways.
D'Hondt method
The most widely used is the D'Hondt formula, using the divisors 1, 2, 3, 4, etc.^{[2]} This system tends to give larger parties a slightly larger portion of seats than their portion of the electorate, and thus guarantees that a party with a majority of voters will get at least half of the seats.
Webster/SainteLaguë method
The Webster/SainteLaguë method divides the number of votes for each party by the odd numbers (1, 3, 5, 7 etc.) and is sometimes considered more proportional than D'Hondt in terms of a comparison between a party's share of the total vote and its share of the seat allocation. This system can favour smaller parties over larger parties and so encourage splits. Dividing the votes numbers by 0.5, 1.5, 2.5, 3.5 etc. yields the same result.
The Webster/SainteLaguë method is sometimes modified by increasing the first divisor to e.g. 1.4, to discourage very small parties gaining their first seat "too cheaply".
Imperiali
Another highest average method is called Imperiali (not to be confused with the Imperiali quota which is a Largest remainder method). The divisors are 1, 1.5, 2, 2.5, 3, 3.5 and so on. It is designed to disfavor the smallest parties, akin to a "cutoff", and is used only in Belgian municipal elections. This method (unlike other listed methods) is not strictly proportional, if a perfectly proportional allocation exists, it is not guaranteed to find it.
Huntington–Hill method
In the Huntington–Hill method, the divisors are given by , which makes sense only if every party is guaranteed at least one seat: this effect can be achieved by disqualifying parties receiving fewer votes than a specified quota. This method is used for allotting seats in the US House of Representatives among the states.
Danish method
The Danish method is used in Danish elections to allocate each party's compensatory seats (or levelling seats) at the electoral province level to individual multimember constituencies. It divides the number of votes received by a party in a multimember constituency by the divisors growing by step equal to 3 (1, 4, 7, 10, etc.). Alternatively, dividing the votes numbers by 0.33, 1.33, 2.33, 3.33 etc. yields the same result. This system purposely attempts to allocate seats equally rather than proportionately.^{[3]}
Quota system
In addition to the procedure above, highest averages methods can be conceived of in a different way. For an election, a quota is calculated, usually the total number of votes cast divided by the number of seats to be allocated (the Hare quota). Parties are then allocated seats by determining how many quotas they have won, by dividing their vote totals by the quota. Where a party wins a fraction of a quota, this can be rounded down or rounded to the nearest whole number. Rounding down is equivalent to using the D'Hondt method, while rounding to the nearest whole number is equivalent to the SainteLaguë method. However, because of the rounding, this will not necessarily result in the desired number of seats being filled. In that case, the quota may be adjusted up or down until the number of seats after rounding is equal to the desired number.
The tables used in the D'Hondt or SainteLaguë methods can then be viewed as calculating the highest quota possible to round off to a given number of seats. For example, the quotient which wins the first seat in a D'Hondt calculation is the highest quota possible to have one party's vote, when rounded down, be greater than 1 quota and thus allocate 1 seat. The quotient for the second round is the highest divisor possible to have a total of 2 seats allocated, and so on.
Comparison between the D'Hondt, SainteLaguë and Huntington–Hill methods
D'Hondt, SainteLaguë and HuntingtonHill allow different strategies by parties looking to maximize their seat allocation. D'Hondt and Huntington–Hill can favor the merging of parties, while SainteLaguë can favor splitting parties (modified SaintLaguë reduces the splitting advantage).
Examples
In these examples, under D'Hondt and Huntington–Hill the Yellows and Greens combined would gain an additional seat if they merged, while under SainteLaguë the Yellows would gain if they split into six lists with about 7,833 votes each.
The total vote is 100,000. There are 10 seats. The Huntington–Hill method threshold is 10,000, which is 1/10 of the total vote.
D'Hondt method  SainteLaguë method (unmodified)  SainteLaguë method (modified)  Huntington–Hill method  

party  Yellow  White  Red  Green  Blue  Pink  Yellow  White  Red  Green  Blue  Pink  Yellow  White  Red  Green  Blue  Pink  Yellow  White  Red  Green  Blue  Pink  
votes  47,000  16,000  15,900  12,000  6,000  3,100  47,000  16,000  15,900  12,000  6,000  3,100  47,000  16,000  15,900  12,000  6,000  3,100  47,000  16,000  15,900  12,000  6,000  3,100  
votes/seat  9,400  8,000  7,950  12,000  11,750  8,000  7,950  12,000  6,000  9,400  8,000  7,950  12,000  7,833  8,000  15,900  12,000  
mandate  quotient  
1  47,000  16,000  15,900  12,000  6,000  3,100  47,000  16,000  15,900  12,000  6,000  3,100  33,571  11,429  11,357  8,571  4,286  2,214  33,234  11,314  11,243  8,485  Disqualified  
2  23,500  8,000  7,950  6,000  3,000  1,550  15,667  5,333  5,300  4,000  2,000  1,033  15,667  5,333  5,300  4,000  2,000  1,033  19,187  6,531  6,491  4,898  
3  15,667  5,333  5,300  4,000  2,000  1,033  9,400  3,200  3,180  2,400  1,200  620  9,400  3,200  3,180  2,400  1,200  620  13,567  4,618  4,589  3,464  
4  11,750  4,000  3,975  3,000  1,500  775  6,714  2,857  2,271  1,714  875  443  6,714  2,857  2,271  1,714  875  443  10,509  3,577  3,555  2,683  
5  9,400  3,200  3,180  2,400  1,200  620  5,222  1,778  1,767  1,333  667  333  5,222  1,778  1,767  1,333  667  333  8,580  2,921  2,902  2,190  
6  7,833  2,667  2,650  2,000  1,000  517  4,273  1,454  1,445  1,091  545  282  4,273  1,454  1,445  1,091  545  282  7,252  2,468  2,453  1,851  
seat  seat allocation  
1  47,000  47,000  33,571  33,234  Disqualified  
2  23,500  16,000  15,667  21,019  
3  16,000  15,900  11,429  14,863  
4  15,900  15,667  11,357  11,399  
5  15,667  12,000  9,400  11,314  
6  12,000  9,400  8,571  11243  
7  11,750  6,714  6,714  9217  
8  9,400  6,000  5,333  8485  
9  8,000  5,333  5,300  7727  
10  7,950  5,300  5,222  7155 
References
 ^ Norris, Pippa (2004). Electoral Engineering: Voting Rules and Political Behavior. Cambridge University Press. p. 51. ISBN 0521829771.
 ^ Gallagher, Michael (1991). "Proportionality, disproportionality and electoral systems" (PDF). Electoral Studies. 10 (1). doi:10.1016/02613794(91)90004C. Archived from the original (pdf) on 4 March 2016. Retrieved 30 January 2016.
 ^ "The Parliamentary Electoral System in Denmark".